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ANGULAR  APERTURE 


OF  OBJECTIVES 


FOR  THE  MICROSCOPE. 


BEAD  BEFOBE  THE  MICBOSCOPICAL  CONGBESS,  AT  INDIANAPOLIS, 
IND.,  AUGUST  15th,  1878, 

BY 

GEO.  E.  BLACKKAM,  M.D.,  F.RM.S, 

President  Dunkirk  (N.  Y.)  Microscopical  Society,  Etc.,  Etc. 


NEW  YORK: 

THE  INDUSTRIAL  PUBLICATION  COMPANY. 

1880. 


THE  GETTY  CENTER 
LIBRARY 


EXPLANATORY  NOTE. 


To  those  readers  who  are  familiar  with  mathematics  and  optics,  it  will 
-doubtless  appear  strange  that  in  the  following  pages  no  use  has  been  made  of 
mathematical  formulae. 

There  are,  however,  a  large  number  of  microscope  users  to  whom  a  purely 
mathematical  discussion  would  be  either  unintelligible,  or  uninteresting,  but 
who  are,  nevertheless,  desirous  of  information  upon  the  much-talked-of  sub- 
ject of  Angular  Aperture. 

It  is  hoped  that  by  the  intentional  selection  of  untechnical  methods  of  dis- 
cussion, and  the  liberal  use  of  carefully  calculated  and  accurately  drawn 
diagrams,  this  paper  has  been  made  interesting  and  useful  to  non-mathemati- 
cal readers  without  entirely  losing  its  value  for  those  who  would  have  pre- 
fered  the  greater  conciseness  of  a  purely  mathematical  discussion. 


2. 


ON  ANGULAR  APERTURE  OF 

OBJECTIVES  FOR  THE  MICROSCOPE. 


N  the  newspaper  report  of  a  recent  popular  lecture  on 
the  microscope,  I  found  the  following  excellent  statement 
of  the  primary  function  of  the  object-glass  of  a  telescope, 
viz. :  "  Thus  we  see  how  this  piece  of  glass,  so  shaped 
and  polished,  gathers  up  the  otherwise  diffused  and  lost 
rays  of  light  that  issue  from  these  distant  objects.  It 
collects  at  least  a  thousand  times  the  quantity  of  light 
that  the  unaided  eye  could  seize,  and  brings  the  whole 
rescued  bundle  of  rays  to  a  focus,  in  which  the  image  of 
the  source  from  which  they  stream  is  brilliantly  repro- 
duced. This  image  the  eye  can  get  near,  and  by  the  aid 
of  a  magnifying  lens  examine." 
I  am  fully  aware  that  newspaper  reports  of  scientific  statements  are  not 
always  to  be  relied  upon;  but  I  hope  that  in  this  case  the  eminent  lecturer 
has  been  reported  correctly,  for  as  it  stands  the  passage  I  have  quoted  is  a  most 
excellent  statement  of  the  principal  function  of  the  object-glass  of  a  telescope,  and 
by  simply  leaving  out  the  word  distant  it  is  equally  true  of  the  object-glass  of  a 
microscope.  Now  the  angular  difference  between  the  paths  of  the  most  divergent 
rays,  which  any  lens  can  thus  gather  up  and  bring  to  a  focus,  is  known  as  the 
.angular  aperture  of  the  lens,  and  forms  the  subject  of  our  paper  this  evening.  It 
would  seem  reasonable  to  suppose  that  if  the  value  of  an  object-glass  depends 
upon  gathering  up  and  bringing  to  a  focus  rays  which  would  otherwise  fail  to 
enter  the  eye,  and  thus  be  dispersed  and  lost,  that  the  more  of  these  rays  it  could 


6 


ON  ANGULAR  APERTURE  OF 


so  utilize,  or,  in  other  words,  the  wider  its  angulur  aperture,  the  better  the  lens ; 
and  this,  indeed,  is  the  fact,  though  many  microscopists,  and  among  them  the 
lecturer  whom  I  have  quoted,  do  not  admit  it. 

Light  is  dispersed  from  every  point  on  the  surface  of  an  object  in  every 
direction  up  to  1800  ;  but,  unaided,  the  human  eye,  according  to  Dr.  David  Brewster, 
is  competent  to  receive  only  a  narrow  pencil  of  io°.  In  other  words,  it  has  an 
angular  aperture  of  only  io°,  and  can  utilize  only  about  1-324  part  of  the  light  ema- 
nating from  the  surface  of  an  object,  the  other  323-324,  or  a  pencil  of  850  on  each 
side,  being  lost  to  it.  It  is  the  problem  of  the  optician  to  gather  up  and  bring  to  a 
focus  as  many  of  these  lost  rays  as  possible.  And  here  let  me  emphasize  the  fact  that 
it  is  only  those  rays  which  are  brought  to  one  common  focus  which  are  of  value,  and 
which  should  be  counted  in  measuring  the  angular  aperture  of  an  objective.  If 
rays  are  admitted  more  divergent  than  can  be  brought  to  a  common  focus,  and  so 
made  to  contribute  to  the  formation  of  the  new  image,  then  those  rays  are  simply 
detrimental,  and  should  be  cut  off  by  means  of  diaphragms. 

Now  if  the  statement  of  the  lecturer  whom  I  have  quoted,  viz. :  "  That  it  is 
the  function  of  the  objective  to  collect  and  bring  to  a  focus  rays  of  light  from  an 
object  too  divergent  to  be  received  by  the  unaided  eye  " — be  correct,  and  my 
corollary  from  that,  "  that  the  more  of  these  lost  rays  that  a  given  glass  can  so 
collect  and  bring  to  a  focus,  the  better  the  glass,"  be  also  correct,  one  would 
naturally  expect  to  find  that  the  improvement  or  evolution  of  the  microscope  was 
accompanied  pari  passu  by  an  increase  of  the  angular  aperture  of  the  objectives  ; 
and  this,  indeed,  we  find  to  be  the  case. 

When,  in  1824,  Mr.  Tulley,  of  London,  produced  the  first  achromatic  micro- 
scope objective  made  in  England  (a  single  combination  of  three  lenses  acting  as 
one),  he  obtained  an  aperture  of  180.  This  seems  small  to  us  now,  but  in  reality 
it  was  a  great  advance,  for  it  is  nearly  double  the  pencil  which  can  be  received  by 
the  unaided  eye. 

Soon  after  the  same  eminent  maker  improved  upon  this  by  adding  another 
combination  in  front,  thus  making  the  first  English  compound  objective,  and  again 
doubled  the  aperture,  getting,  with  the  double  combination,  an  aperture  of  38 °. 

In  1829,  Mr.  Joseph  Jackson  Lister,  published  his  celebrated  paper,  showing 
how  many  of  the  difficulties  which  had  interfered  with  the  use  of  two  or  more 
combinations  together  could  be  overcome,  and  exhibited,  in  confirmation  of  his 
conclusions,  an  objective,  of  which  it  is  recorded  that  it  "  gave  a  large  and  correct 
field,  and  transmitted  a  pencil  of  500."  This  was  indeed  progress;  but  the  end 
was  not  yet.  In  1837,  Mr.  Thomas  Ross,  an  eminent  London  optician,  pre- 
sented a  paper  to  the  Society  of  Arts,  detailing  his  discovery  of  the  negative  aber- 
ration produced  by  the  cover  glass,  and  the  means  he  had  devised  to  neutralize 
it  by  approximating  the  front  and  middle  combinations  of  the  objective.  In  this 
paper,  Mr.  Ross  states  that  he  has  made  an  improved  combination,  of  which  he 


OBJECTIVES  FOR  THE  MICROSCOPE, 


7 


says  :  "  The  focal  length  is  y&  inch,  having  an  angular  aperture  of  6o°,  with  a 
distance  of  1-25  of  an  inch.  Later  he  announced  that  "  on  several  occasions  the 
enormous  angle  of  1350  had  been  obtained,"  and  unfortunately  added  that  "  1350 
is  the  largest  angular  pencil  that  can  be  passed  through  a  microscope  object-glass." 
Chas.  A.  Spencer,  the  now  famous  father  of  American  microscope  making,  was 
led  to  a  theoretical  and  practical  examination  of  the  validity  of  this  statement. 
The  supposed  theoretical  grounds  of  the  assumption  not  having  been  found  to 
sustain  Mr.  Ross's  position,  conclusive  evidence  of  its  incorrectness  was  speedily 
obtained  by  the  construction  of  a  1-12  inch  objective,  having  an  angle  of  aperture 
of  1460.  And  in  the  catalogue  of  Ross  &  Co.,  for  November,  1874, 1  find  advertised 
"Ross'  New  Patent  Object  Glasses.  Devised  by  Mr.  Wenham."  i-25th  aper- 
ture about  1600.  Mr.  Spencer  claimed  1780  aperture  for  his  1-12  in  185 1,  and  Mr. 
Tolles,  of  Boston,  now  makes  lenses  which  have  air  angles  of  infinitely  near  1800, 
and  immersion  angles  greater  than  that  corresponding  to  this  enormous  air  angle.* 
Thus  we  see  that,  as  might  have  been  expected,  the  record  of  the  gradual  im- 
provement of  the  microscope  is  the  record  of  gradually  increasing  angular  aper- 
ture of  objectives,  till  at  length  the  extreme  limit  of  possible  air  angle,  i.e.,  an 
aperture  only  a  differential  less  than  1800  has  been  reached.  That  these  modern 
wide-angled  lenses  are  better  than  their  narrow-angled  predecessors  or  contempo- 
raries, is  shown  by  the  fact  that  many  objects  entirely  invisible  with  the  narrow- 
angled  lenses  are  clearly  defined  by  wide-angled  lenses  of  less  amplifying  power. 
Among  them  I  may  mention  that  Band  No.  XIX  of  Nobert,  112,594  lines  to  the 
inch,  (each  line  being  only  about  1-225 198  °f  an  mcn  wide),  has  been  clearly  resolved 
with  my  Tolles'  1-6  of  (nearly)  1800  air  angle ;  and  that  the  flagellum  of  Bacteriiwi 
tetmo  was  seen  by  Drysdale  and  Dallinger  with  a  new  wide-angled  ^  of  Powell 
&  Lealand,  when  the  comparatively  narrow-angled  1-50  of  the  same  makers  failed 
to  reveal  the  existence  of  this  tiny  appendage  of  the  pigmy  of  the  bacteria. 

It  has  been  objected  to  wide-angled  lenses  that  they  possessed  less  "penetra- 
ting power"  or  more  properly  less  "depth  of  focus"  than  narrow-angled  lenses.  That 
is  to  say,  that  the  layer  of  an  object  that  could  be  seen  without  change  of  focus,  is 
thinner  with  wide  than  narrow-angled  lenses.  If  this  were  true,  it  would  be  an 
argument  in  favor  of  the  wide-angled  lenses  instead  of  against  them.  In  reality, 
however,  it  does  not  depend  upon  the  aperture,  but  is  only  residual  spherical  aber- 
ration which  can  be  left  in  and  distributed  in  a  wide-angled  lens,  as  well  as  in  a 
narrow-angled  one.  It  is,  however,  as  I  have  said,  only  residual  error  at  best,  and 
the  less  a  lens  has  of  it  the  better  the  lens.  This  will,  I  think,  be  easily  seen  upon 
an  inspection  of  the  diagram  (Fig.  1),  showing  the  action  of  an  uncorrected 
plano-convex  lens  of  crown  glass.  The  rays  from  the  nearer  surface  of  the  object 
which  impinge  upon  the  peripheral  portions  of  the  lens  would,  if  the  lens  were  free 


*  Since  this  was  written,  other  makers,  both  here  and  abroad,  have  made  and  are  making  objectives  whose 
immersion  angles  are  greater  than  that  corresponding  to  (infinitely  near)  i8o°air. 


8 


ON  ANGULAR  APERTURE  OF 


from  spherical  aberration,  be  brought  to  a  focus  further  back  than  those  from  the 
further  surface  of  the  object;  as  it  is.  however,  they  are  both  brought  to  the  same 
focus  by  reason  of  the  spherical  aberration.  Such  a  lens  has  a  good  deal  of  pene- 
trating power  or  depth  of  focus ;  but  its  definition  is  not  satisfactory.  A  com- 
mon bulls-eye  condenser  is  a  good  sample  of  this  kind  of  lens.  The  same  holds 
true  of  all  objectives  possessed  of  penetrating  power,  whatever  their  angular  aper- 
ture. The  only  legitimate  method  of  obtaining  depth  of  focus  or  "  penetration," 
is  by  increasing  the  anterior  conjugate  focus  or  working  distance,  so  that  the  thick- 
ness of  the  layer  it  is  desired  to  see  on  each  side  of  the  true  focal  plane,  may  be 
relatively  small.  Thus  a  one  inch  objective,  with  an  anterior  focus  of  -317  of  an 
inch,  will  bear  amplification  up  to  400  diameters,  and  at  that  power  might 
properly  show,  with  reasonable  clearness,  a  layer  of  the  object  on  each  side  of  the 
true  focal  plane  much  thicker  than  that  which  a  1-5  with  only  -018  of  an- 
terior focus  ought  to  show  at  the  same  amplification.  It  is,  perhaps,  true,  that  by 
skillful  management  the  residual  spherical  aberration  can  be  so  distributed  that 
several  planes  of  an  object  may  be  in  view  at  once,  but  this  is  always  at  the  sac- 
rifice of  definition,  and  as  the  better  the  images  the  more  noticeable  do  errors 
resulting  from  this  plan  of  overlapping  several  of  them  become — wide-angled  lenses 
show  the  defects  of  this  plan  more  markedly  than  narrow-angled  lenses,  whence 
has  arisen  the  fallacy  that  narrow-angled  lenses  are  possessed  of  an  inherent 
property  of  "  penetration,"  and  a  residual  error  has  been  lauded  as  a  virtue. 

This  much  as  to  the  value  of  angular  aperture;  now  for  the  question  "  What 
is  an  angular  aperture  ?  " 

I  have  already  defined  it  as,  The  angular  difference  between  the  paths  of  the 
most  divergent  rays  which  an  objective  can  gather  up  and  bring  to  a  focus. 

Let  us,  however,  examine  some  of  the  standard  authors,  and  see  what  they 
have  to  say  on  the  subject : 

Dr.  W.  B.  Carpenter  ("The  Microscope,"  4th  Ed.,  1868)  says:  "  The  angle 
of  aperture,  that  is,  the  angle  made  by  the  most  diverging  rays  of  the  pencil 
issuing  from  any  point  of  an  object  that  can  enter  the  lens." 

Prof.  L.  Beale  ("  How  to  Work  with  the  Microscope,"  4th  Ed.,  1870) :  "  The 
angle  of  aperture  is  the  angle  made  by  two  lines  from  opposite  sides  of  the  aper- 
ture of  the  object-glass  with  the  point  of  focus  of  the  lens." 

Dr.  H.  Frey  ("  The  Microscope  and  Microscopical  Technology  " ;  Cutter's 
Translation;  1872):  "  The  term,  angle  of  aperture  of  a  lens,  denotes  the  angle 
which  is  formed  by  the  focus  and  the  two  terminal  points  of  the  diameter  of  the 
lens." 

Dr.  Wythe,  of  San  Francisco  ("  Microscopists  Manual,"  3d  Ed.,  1877): 
"  Angular  Aperture. — The  angle  made  by  the  diameter  of  the  actual  aperture  of 
an  objective,  and  the  distance  from  its  focal  point." 

Mr.  F.  H.  Wenham  is  thus  quoted  and  endorsed  by  Chas.  Brooke,  Prest.  R. 


OBJECTIVES  FOR  THE  MICROSCOPE. 


9 


M.  S.,  in  his  annual  address  to  the  Society,  in  1875:  "Mr.  Wenham  is  un- 
questionably right  in  stating  that  if  an  isosceles  triangle  be  described,  the  base  of 
which  is  ten  times  the  measured  diameter  of  the  front  lens,  and  the  altitude  ten 
times  the  measured  distance  of  the  focal  point  from  the  same  surface,  the  vertical 
angle  ot  that  triangle  will  correctly  represent  the  maximum  available  aperture." 

Now,  strange  as  it  may  seem  to  question  the  concurrent  testimony  of  such 
distinguished  authorities,  I  am  not  disposed  to  accept  any  of  the  definitions  I  have 
quoted,  as  correct.  They  all  lack  accuracy  and  universality  of  application.  To 
illustrate.  r 

Let  me  take  a  hemispherical  lens  of  crown  glass,  whose  index  of  refraction  is 
1*525,  and  radius  of  curvature  is  -015  of  an  inch.  The  diameter  will,  of'.course,  be  , 
•03  of  an  inch,  and  its  principal  focus  for  parallel  rays  '0286  of  an  inch.  As  the 
focal  length  is  measured  from  the  optical  centre,  which,  in  the  case  of  a  plano- 
convex lens,  is  situated  on  the  convex  surface  of  the  lens,  where  it  is  cut  by  the 
optic  axis,  if  we  turn  the  plane  side  of  the  lens  towards  an  object  the  working  dis- 
tance will  of  course  be  the  focal  length  minus  the  thickness  of  the  lens.  In  this 
case  f  I  =  -0286  —  thickness  =  -015  —  working  distance  -0136.  This  gives  us  for  our 
isosceles  triangle  a  base  of  -03  of  an  inch,  and  an  altitude  of  -0136  of  an  inch,  or, 
to  follow  out  Mr.  .Wenham's  plan,  multiplying  by  ioo-  gives  us  a  base  of  3  inches, 
and  an  altitude  of  1*36  inch;  and  on  this  scale  I  have  drawn  the  diagram,  the 
vertical  angle  of  our  triangle  will  be  95 0  36'  (Fig.  2,).  If  our  object  was  placed 
in  the  principal  focus,  however,  the  posterior  conjugate  focus  would  be  infinitely 
distant,  and  in  order  to  get  nearer  to  the  conditions  under  which  an  objective  is 
actually  used,  let  us  bring  our  posterior  conjugate  focus  to  10  inches;  this  will 
lengthen  our  anterior  focus  to  '0157,  nearly;  our  enlarged  triangle  will  then  be 
base  3  inches,  altitude  1-57  inches,  nearly,  and  the  resulting  angle  870  22'  (Fig.  3). 
But,  in  point  of  fact,  the  spherical  aberration  would  be  so  great  that  the  outer  rays 
of  this  pencil  would  be  brought  to  a  focus  at  a  distance  considerably  less,  and 
would  not  enter  into  the  formation  of  the  image  at  10  inches,  but  would  only 
serve  to  confuse  it,  and  would  have  to  be  cut  off  by  a  diaphragm.  If  this  dia- 
phragm were  placed  behind  the  lens,  the  diameter  of  the  front,  and  the  distance 
to  the  focal  point,  and  consequently  our  triangle,  would  remain  unchanged ;  and 
our  triangle  would  consequently  indicate  an  angular  aperture  far  beyond  the  real 
available  angular  aperture  of  the  lens.  This,  too,  would  be  the  case  of  the  lens 
under  consideration  were  the  point  of  a  compound  objective  in  which  the  back 
combinations  were  unable  to  transmit  the  entire  pencil  received  by  the  front,  or,  if 
transmitted,  to  correct  the  aberrations  sufficiently  to  bring  all  the  rays  received  by 
the  front  to  a  common  focus  at  the  eye-piece.  This  is,  in  fact,  the  case  with  many 
objectives  which  have  not  been  properly  corrected ;  they  will  admit  rays  far  more 
divergent  than  the  extreme  pencil  which  they  can  bring  to  a  common  focus  at  the 
eye-piece.     For  these  objectives,  if  used  dry,  Mr.  Wenham's  rule  will  give  an 


IO 


ON  ANGULAR  APERTURE  OF 


aperture  in  excess  of  the  maximum  available  aperture,  thus  giving  undue  credit  to 
faulty  lenses.  It  is  this  class  of  lenses  which  are  improved  by  having  the  aperture 
reduced,  as  detailed  by  Dr.  Piggott,  but  the  argument  is  not,  therefore,  that  all 
wide-angled  lenses  would  be  benefited  by  a  reduction  of  aperture,  but  only  those 
in  which  the  marginal  rays  have  not  been  properly  corrected. 

Now,  in  order  to  understand  the  matter  fully,  it  is  necessary  for  us  to  remem- 
ber that  in  the  vast  majority  of  cases,  objects  viewed  through  the  microscope  are 
seen  under  very  different  conditions  from  those  under  which  we  ordinarily  view 
objects  with  the  naked  eye. 

In  the  case  of  objects  viewed  with  the  naked  eye,  we  see  them  without  the 
interposition  between  them  and  the  eye  of  any  medium  but  air,  the  index  of  re- 
fraction of  which  is  so  small  as  to  be  practically  of  no  effect,  and  consequently 
the  rays  of  light  which  radiate  from  them,  either  primarily  or  by  reflection,  reach 
the  eye  without  appreciable  refraction,  and  from  a  distance  of  not  less  than  about 
eight  or  ten  inches. 

Objects  viewed  through  the  microscope,  on  the  other  hand,  are  frequently 
immersed  in  some  highly  refractive  medium,  such  as  water,  glycerine,  or  Canada 
balsam,  are  generally  covered  with  a  thin  plate  of  glass,  with  parallel  surfaces,  and 
being  more  or  less  transparent  are  viewed  by  rays  of  light  which  pass  through 
them  from  below,  and  so  through  the  cover  glass  to  the  microscope  and  to  the  eye. 

The  importance  of  this  distinction  will  be  seen  when  we  come  to  discuss 
the  question  of  immersion  apertures,  and  especially  apertures  beyond  the  extreme 
limit  possible  for  dry  objectives. 

Let  us  now  take  our  hemispherical  lens  of  crown  glass,  as  before,  and  sup- 
pose it  to  be  the  front  lens  of  an  objective  whose  posterior  combinations  are  so 
arranged  as  to  leave  its  anterior  focal  length  unchanged  while  correcting  its  aber- 
rations, and  thus  bringing  all  the  rays  that  can  enter  it  into  a  common  focus  at  the 
eye-piece.  Allow  -002  of  an  inch  for  the  setting ;  we  should  still  have  -028  of  an 
inch  available  front ;  our  enlarged  triangle  will  then  have  for  dimensions :  base 
2-80,  altitude  1*57,  vertical  angle  830  26',  nearly,  (Fig.  4).  This,  then,  would 
be  the  angular  aperture  of  that  objective  under  those  circumstances. 

Of  course,  if  the  radiant  point  were  placed  closer  to  the  lens  than  its  principal 
focus,  the  angle  made  by  the  extreme  rays  which  entered  the  face  of  the  glass 
would  be  increased,  but  in  that  case  they  would  be  still  somewhat  divergent  on 
emerging  from  the  posterior  side  of  the  front  lens,  and  without  further  refraction 
would  not  meet  at  any  posterior  conjugate  focus,  and  of  course  form  no  image. 
This  further  refraction  is  afforded  by  the  posterior  combinations  of  the  compound 
objective,  and  as  a  rule  the  actual  focal  length  of  a  compound  objective  is  much 
less  than  that  of  its  front  lens  taken  alone;  and  this  focal  length  of  the  objective, 
as  a  whole,  varies  with  the  distance  of  the  posterior  conjugate  focus  at  the  eye- 
piece ;  that  is  to  say,  with  the  length  of  the  tube  of  the  microscope,  and  also  with 


OBJECTIVES  FOR  THE  MICROSCOPE. 


I  I 


the  distance  between  the  front  lens  and  the  middle  combination,  or  lens,  of  the  ob- 
jective— a  distance  which  is  variable  in  objectives  having  correction  for  thickness  of 
covering  glass. 

It  is  quite  possible,  however,  that  the  posterior  combinations  might  not  be 
capable  of  transmitting  the  most  divergent  rays  which  could  pass  through  the  front, 
or  even  if  capable  of  transmitting  them  might  not  be  capable  of  correcting  their 
aberrations  (which  are  always  the  greatest  for  marginal  rays).  In  either  case  the 
rule  propounded  by  Mr.  Wenham,  and  endorsed  by  President  Brooke,  to  which  I 
have  before  referred,  would  give  inaccurate  results,  and  the  vertical  angle  of  an 
isosceles  triangle  whose  base  was  ten  times  the  measured  diameter  of  the  front  lens, 
and  the  altitude  ten  times  the  distance  of  the  focal  point,  would  not  correctly  repre- 
sent the  maximum  available  aperture,  but  something  in  excess  thereof.  In  fact,  I 
think  it  is  demonstrable  that  this  rule  can  never  give  the  maximum  available  aper- 
ture of  an  objective. 

Mr.  Wenham,  himself,  seems  to  have  become  conscious  of  this,  for  he  has 
since  proposed  other  methods  and  modifications,  each  in  turn  announced  to  be 
the  only  reliable  one,  till  the  student  who  has  tried  to  follow  him  through  his  dis- 
cussion of  this  matter  gets  lost  among  his  numerous  amendments.  One  method 
of  his,  set  forth  in  the  London  Monthly  Microscopical  Journal  for  March,  1874,  is 
to  place  in  the  focus  of  the  microscope  a  slide,  the  upper  surface -of  which  is 
covered  with  some  opaque  material  (he  suggests  platinum  foil  here),  through  which 
a  slit  is  cut,  the  edges  of  which  serve  to  cut  off  extraneous  rays,  and  then  take  the 
aperture  in  the  usual  way  with  a  sector :  that  is,  by  placing  a  light  in  front,  and 
either  rotating  the  microscope  around  the  object  as  centre  (as  can  be  done 
with  Beck's,  Zentmayer's  and  Tolles'  largest  stands),  till  the  light  disappears 
from  the  centre  of  the  field,  or  by  making  the  light  traverse  the  circumference  of 
a  circle  of  which  the  object  is  the  centre  (as  can  be  done  with  the  stand  made  for 
me  by  Mr.  Tolles)  till  the  same  result  occurs.  This  will  give  one-half  the  angle, 
and  of  course  multiplying  by  two  will  give  the  entire  angular  aperture.  In  this 
paper  (March,  1874)  Mr.  Wenham  says,  "it  is  preferable  to  open  the  slit  till  the 
edges  appear  in  the  margin  of  the  field."  He  changed  his  mind  about  this  after- 
wards, and  stated,  "  the  narrower  the  slit  the  more  accurate  the  result  will  be."  I 
can  not  give  date  and  page  for  this,  but  he  quotes  it  himself  in  the  Monthly  Mi- 
croscopical Journal  for  December,  1876,  and  adds  :  "  This  means  strictly  that  for 
absolute  accuracy  we  must  approach  to  a  line  and  cut  off  all  rays  in  the  focal 
plane  on  either  side,  quite  up  to  the  axis  of  the  object-glass." 

I  may  here  say  that  this  plan  has  some  of  the  faults  of  his  old  triangle 
method ;  it  will  give  the  most  oblique  ray  that  can  enter  the  lens  from  the  ob- 
ject, but  it  will  not  give  a  clear  indication  whether  such  rays  can  be  utilized  to 
produce  a  well-defined  image  of  the  object.  I  know  of  object-glasses  that 
give  by  this  method  a  very  large  angle,  but  from  lack  of  accurate  correction  for 


12  ON  ANGULAR  APERTURE  OF 

i 

the  very  oblique  rays,  their  effective  angle  is  much  less  than  the  one  indicated 
by  this  plan. 

This  difficulty  also  occurred  to  Mr.  Wenham,  and  he  moved  another  amend- 
ment on  himself,  so  he  gives  the  diagram  which  I  have  copied  here  (Fig.  5),  and 
the  following  description  :  "  I  now  adopt  the  following  method  of  measuring 
apertures:  a  is  the  working  diameter  of  an  object-glass;  b  the  central  pencil  or 
true  angle  of  aperture ;  c  c,  oblique  or  lateral  pencils  enclosing  the  field  of  view ; 
d  d,  a  slit  of  considerable  width,  with  parallel  edges  attached  to  a  glass  slip,  e. 
In  order  to  measure  apertures,  the  object-glass  is  first  adjusted  and  focused  on 
the  upper  surface  of  the  glass  slip.  One  edge  of  the  slit  is  now  brought  forward 
so  as  to  exactly  bisect  the  field  of  view,  half  of  which  will  appear  quite  dark. 
Over  the  eye-piece  is  now  placed  a  cap  containing  a  biconcave  lens  of  about  half 
an  inch  radii ;  by  means  of  this  and  the  movement  of  the  sliding  containing  tube, 
a  distinct  telescopic  image  of  a  distant  lamp,  or  other  bright  object,  may  be  obtained 
through  the  open  half  of  the  object-glass.  Turn  the  open  end  away  from  the 
lamp  by  rotating  the  microscope,  and  the  flame  will  suddenly  disappear  at  the 
point  where  it  is  observed  by  the  edge  of  the  slit.  Mark  this  as  zero !  Now  re- 
move the  lens  from  over  the  eye-piece,  bring  back  the  slit  till  the  opposite  edge 
obscures  the  other  half  of  the  field,  and  again  exactly  bisects  it,  seeing  that  the 
plane,  <?,  is  still  in  focus ;  replace  the  cap  and  turn  the  microscope  till  the  flame 
again  vanishes,  and  true  aperture  will  be  indicated." 

In  the  last  issue  of  the  Monthly  Microscopical  Journal  (Nov.  and  Dec,  1877), 
Mr.  Wenham  has  changed  his  plan  again;  he  has  abandoned  his  slit  in  the 
focus  of  the  objective,  but  still  uses  the  microscope  as  a  telescope.  He  says : 
"  The  arrangement  that  I  now  make  use  of  consists  of  an  '  examining  lens '  placed 
over  the  lowest  eye-piece.  This  lens  is  a  plano-convex  achromatic  of  near  four- 
tenths  of  an  inch  focus,  contained  in  a  tube  sliding  in  an  outer  one,  firmly  fitting 
on  to  the  eye-piece  nozzle ;  at  a  distance  of  one  and  a  half  inches  behind  the  lens 
there  is  a  removable  cap,  containing  a  thin  plate  with  a  central  stop  (he  means  a 
hole)  of  one-fiftieth  of  an  inch  in  diameter.  The  small  size  of  this  stop,  and  the 
distance  that  it  is  placed  from  the  lens,  ensures  the  fixed  direction  of  the  eye  in 
the  axis,  and  prevents  any  rays,  except  those  of  the  central  pencil,  from  entering. 
By  means  of  the  draw-tube,  a  lamp  flame,  or  other  object  taken  for  an  index  is 
focused  for  distinct  vision  without  the  stop.  Replace  this  and  take  the  angle  of 
the  objective,  either  by  rotating  on  a  sector  in  the  usual  way,  or  by  measuring 
the  angle  between  two  objects  set  the  requisite  distances  asunder,  the  apex  being 
at  the  focal  point  of  the  object-glass." 

To  these  last  two  methods  it  might  be  very  well  objected  that  most  of  the 
conditions  under  which  a  microscope  is  used  in  practice  are  here  reversed ;  that 
distant  objects  are  viewed  instead  of  near  ones ;  diminished  images  seen  instead 
of  enlarged  ones ;  that  no  account  is  made  of  the  effects  of  cover  glass  or  cover 


OBJECTIVES   FOR  THE  MICROSCOPE. 


13 


correction,  and  that  it  is  manifestly  absurd  and  unscientific  to  use  a  microscope  as. 
a  telescope-  in  order  to  determine  its  qualities  (or  one  of  them — angular  aperture) 
when  used  as  a  microscope ;  but  instead,  we  will  let  Mr.  Wenham  answer  himself. 
In  the  Monthly  Microscopical  Journal \  for  November,  1872,  Mr.  Wenham  says: 
"  Prof.  Govin  has  proposed  to  associate  the  measurement  of  the  angle  of  aperture 
with  the  simultaneous  view  of  an  object  distinctly  defined,  like  the  flames  of  two 
candles  placed  asunder,  or  two  white  strips  separated  on  a  black  screen,  to  the 
limit  of  distinct  visibility ;  the  angle  from  these  two  points  to  the  focus  of  the  ob- 
ject-glass will  represent  the  aperture.     The  microscope  is  thus  converted  into  a 
kind  of  telescope  by  means  of  a  pair  of  lenses  over  the  eye-piece,  similar  to  what  is 
known  as  Ross's  '  examining  glass.'     Unfortunately  for  the  success  of  this  plaiy 
different  optical  combinations  at  the  eye-piece  give  different  results  by  elongating 
or  shortening  the  conjugate  focus."    In  view  of  this  statement  of  Mr.t  Wenham  . 
himself,  of  the  effect  of  different  optical  combinations  at  the  eye-piece  giving  dif- 
ferent results,  the  consistency  of  his  proposition  in  1876  to  use  &\  bicdjic\iv£  lens 
of  about  half  an  inch  radii"  over  the  eye-p:.ece,  and  in  1877  to  use  "  a  piano 
convex  lens  of  near  four-tenths  of  an  inch  focus  "  over  the  eye-piece,  to  obtain  the 
same  result,  becomes  beautifully  apparent. 

My  only  apology  for  taking  up  so  much  time  in  quoting  the  various  con- 
flicting dogmas  promulgated  by  Mr.  Wenham  in  regard  to  the  measurement  of 
angular  aperture,  is  that  he  has  been  the  most  prominent  participator  on  one  side 
of  this  discussion,  which  has  been  carried  on  for  several  years ;  and  as  he  is  still 
quoted  by  his  admirers  as  an  authority  on  the  subject,  it  seemed  best  to  quote 
liberally  from  his  contributions  to  the  confusion  of  knowledge  on  this  subject,  in 
order  to  show,  by  his  own  writings,  the  condition  of  self-contradiction  and  general 
absurdity  to  which  he  has  been  reduced  by  his  successive  attempts  to  overthrow 
Mr.  Tolles'  claim  to  having  constructed  objectives  of  extreme  angle ;  and  to  thereby 
demonstrate  the  total  untrustworthiness  of  Mr.  Wenham  as  an  authority  on  angular 
aperture,  however  eminent  he  may  be  as  an  inventor  of  binocular  arrangements, 
reflex  illuminators,  and  "  patent  "  objectives. 

The  question  then  arises,  "  How  should  angle  of  aperture  be  measured  ?  "  and 
the  answer  is  that  the  angle  of  aperture  being  the  angular  distance  between  the 
extreme  rays  of  the  widest  pencil  which  the  objective  can  gather  into  one  common- 
focus,  with  the  production  of  a  well-defined  image  at  the  eye-piece,  it  is  necessary* 
to  measure  the  angle  of  the  objective  when  in  actual  use  on  the  microscope,  with, 
an  object  in  the  centre  of  the  field,  and  giving,  in  conjunction  with  the  eye-piece, 
the  most  perfect  definition  possible.  If  now  we  can  measure  the  angle  contained 
between  the  most  oblique  ray  of  the  pencil  actually  utilized  in  the  production  of  a 
well-defined  image,  and  the  optical  axis  of  the  instrument,  it  will  be  just  one-half 
the  available  aperture  of  the  lens  j  and  by  simply  doubling  it  we  will,  of  course, 


14 


ON  ANGULAR  APERTURE  OF 


obtain  the  full  available  aperture.*  We  will  first  consider  the  case  of  an  objective 
when  used  on  an  object  uncovered  in  air. 

Objects,  when  examined  in  the  microscope,  are  usually  placed  upon  slides  of 
crown  glass,  whose  surfaces  are  parallel  to  each  other  and  at  right  angles  to  the 
optic  axis  of  the  microscope. 

I  have  such  a  one  upon  the  stage  of  my  microscope,  and  Fig.  6  represents 
the  same  on  an  enlarged  scale.  The  light  enters  from  below,  is  refracted  toward 
the  axis  in  the  body  of  the  slide,  and  the  surfaces  of  the  slide  being  parallel,  and 
the  ray  being  incident  from  and  emergent  into  the  same  medium  (air),  it  follows, 
from  a  well-known  optical  law,  that  the  angle  formed  by  it  with  the  normal  is  the 
same  on  both  sides  of  the  slide ;  but  the  optic  axis  of  the  microscope  is  in  this 
case  the  normal,  whence  it  follows  that  if  we  can  measure  the  angle  of  incidence 
below  the  slide,  or  in  other  words,  the  angle  of  illumination,  we  shall  obtain  the 
angle  of  emergence,  which,  when  multiplied  by  two,  will  give  us  the  angle  of  the 
lens. 

I  have  here  a  Student  }(,  sent  me  by  Mr.  Tolles,  and  stated  by  him  to  have 
aperture  of  about  no°.  The  diameter  of  the  exposed  surface  of  its  front  lens  is 
o-i6  of  an  inch,  and  its  working  distance,  when  used  uncovered  with  the  j4  in-  solid 
eye-piece  on  this  stand  is  '015,  which  gives  for  the  vertical  angle  of  Mr.  Wenham's 
triangle  1580  46',  nearly,  (Fig.  7).  If,  however,  we  take  the  diameter  of  the  light 
spot  seen  when  looking  through  the  front  of  the  lens,  that  is  the  clear  aperture, 
we  will  find  it  to  be  '077,  which,  with  the  same  frontal  distance,  will  give  us  about 
1370  26'  for  our  vertical  angle  (Fig.  8).  Each  of  these  is  so  largely  in  excess 
of  the  angle  claimed  by  the  maker  as  to  suggest  a  doubt  of  the  correctness  of 
either,  and  we  will  proceed  to  actual  measurement.  Attached  to  my  stand  is  a 
graduated  circle,  whose  centre  is  in  the  horizontal  plane  of  the  object ;  on  this 
circle  rides  a  fitting  carrying  mirrors  and  accessory  holder,  and  an  index  which 
stands  at  zero  when  the  centre  of  the  fitting  is  in  the  optic  axis  of  the  instrument. 
Removing  the  accessory  holder,  we  use  its  socket  as  a  candlestick,  to  hold  a  toy 
candle,  and  turn  the  microscope  so  that  its  body  is  horizontal,  and  get  from  our 
tiny  candle  flame  central  illumination.  Now,  turning  the  holder  in  the  graduated 
circle,  we  swing  the  light  around  our  object  as  a  centre  till  either  the  image  be- 
comes imperfect  or  the  centre  of  the  field  darkened,  and  we  get  one-half  the  useful 
aperture  of  the  lens  in  this  condition,  which  is  500,  making  the  total  aperture 
at  uncovered,  ioo°.    (Fig.  9). 

There  is  one  objection  to  this  plan,  which  is  that,  on  account  of  the  refraction 
at  the  lower  surface,  the  ray  does  not  proceed  directly  from  the  lamp  to  object. 
This  objection  is  valid,  but  resulting  error  is  very  small,  on  account  of  the  small 


*  In  practice  it  is  much  better  to  take  two  readings — one  right  and  one  left  of  the  optic  axis  of  the 
microscope — and  add  them  together  to  obtain  total  angle  of  aperture,  as  by  this  method  any  errors  arising  from  lack 
of  absolute  accuracy  in  centering  the  illumination  are  eliminated. 


OBJECTIVES  FOR  THE  MICROSCOPE. 


thickness  of  the  slide ;  but  even  this  slight  error  will  be  eliminated  by  the  method 
which  I  shall  finally  propose. 

Let  us  now  take  the  case  of  an  object  mounted  dry  under  a  glass  cover. 

I  have  such  a  one  here.  A  slide  is  rendered  opaque  by  a  coating  of  photo- 
graphers' collodion,  which  has  been  blackened  by  exposure  to  light.  In  this  a 
narrow  slit,  1-300  wide,  has  been  cut  with  a  needle  point,  and  diatoms  have  been 
mounted  on  the  slide,  so  that  some  of  them  will  lie  in  the  slit,  and  one  portion 
flooded  with  balsam,  and  the  rest  left  dry,  and  all  covered  with  a  disc  of  glass  just 
•009  inch  thick.  We  now  adjust  our  quarter,  with  same  eye-piece,  over  one  of  the 
diatoms  on  the  dry  part,  finding  it  necessary  to  close  the  combinations  very  con- 
siderably from  their  adjustment  for  uncovered.  This  does  not  change  the  diame- 
ter of  our  front,  but  it  does  change  the  angular  aperture  of  the  lens.  We  now 
have  a  working  distance  of  "009,  which,  plus  thickness  of  cover  glass,  "009,  and 
air  space,  -ooi,  gives  frontal  distance  of  -019.  Taking  the  diameter  of  light  spot 
of  our  lens  for  a  base,  and  "019  for  vertical  light  of  our  isosceles  triangle,  Mr. 
Wenham's  rule  gives  us  1270  28'  (Fig.  10)  for  angle;  if  we  take  the  exposed 
diameter  of  our  front  lens  for  base  the  angle  is  1530  16'.  (Fig.  11).  Measuring 
once  again  by  swinging  our  candle  around  the  object,  we  find  that  good  definition 
ceases  at  about  550  from  axis,  giving  no°  as  available  angle  of  our  lens  when 
adjusted  for  cover  9-1000  inch  thick.  This  is  more  than  the  angle  when  adjusted 
for  uncovered,  though  the  frontal  distance  is  larger,  the  increase  being  due. to  the 
change  in  the  relative  positions  of  the  combinations  of  the  objective  itself. 

Tracing  now  our  ray  from  the  candle  to  objective  (Fig.  12),  we  find  it  incident 
from  air  to  lower  surface  of  slide  at  550,  refracted  in  the  glass  to  320  30',  emergent 
into  air  below  the  cover  at  550,  at  which  it  is  incident  to  lower  surface  of  cover;  re- 
fracted in  cover  to  320  30',  and  finally  emergent  into  air  at  550.  It  is  impossible 
to  show  graphically  the  deviation  of  the  ray  in  the  tiny  air  space  of  1-1000 
of  an  inch  in  thickness,  without  drawing  diagrams  on  an  enormous  and  un- 
wieldly  scale — this  tiny  air  space  is  an  important  factor,  however,  as  we  shall  here- 
after see. 

I  have  used  the  slit  here,  not  because  it  is  of  any  real  importance,  but  to 
approximate  my  method  to  one  of  Mr.  Wenham's,  and  because  it  affords  a  con- 
venient method  of  ascertaining  when  the  object  is  in  the  centre  of  the  field.  If 
our  object  is  immersed  in  balsam,  and  covered,  the  result  will  be  practically  un- 
changed, except  that  the  ray  of  light  will  suffer  no  appreciable  refraction  between 
the  object  and  cover,  but  will  pass  in  a  straight  line  through  the  slide,  balsam,  and 
cover,  and  have  its  final  emergence  into  air,  the  same  as  its  incidence,  below  the  slide. 
In  the  remainder  of  this  paper  I  shall  treat  of  the  object  considered  as  mounted 
in  balsam  and  covered  with  a  cover  which,  together  with  the  balsam,  shall  be  -oio 
inch  thick,  unless  where  something  different  is  specially  stated. 

It  is  evident  that  in  order  to  increase  the  angle  of  aperture  of  our  dry 


i6 


ON  ANGULAR  APERTURE  OF 


lens,  we  must  either  increase  the  diameter  of  that  portion  of  the  front  lens,  which 
is  actually  used,  or  must  shorten  the  frontal  distance,  and  in  either  case  the  pos- 
terior combination  of  the  objective  must  be  correspondingly  modified  so  as  to 
transmit  and  bring  to  a  common  focus  all  the  rays  of  the  wider  pencil  thus  ad- 
mitted by  the  front.  In  practice  it  will  be  found  that  the  increase  of  angle  in 
dry  lenses  is  obtained  by  shortening  the  frontal  distance  so  that,  with  dry  lenses  of 
very  wide  angle,  say  from  1500  up  to  nearly  1800,  the  covering-glass  must  be  very 
thin,  and  the  front  of  the  objective  almost  in  contact  with  the  cover.  The  dif- 
ficulty and  inconvenience  resulting  are  well  known,  and  have  done  much  to  create 
the  prejudice  which  exists  in  the  minds  of  many  of  our  older  microscopists  against 
wide-angled  objectives. 

Let  us  now  consider  again  the  case  of  an  object  immersed  in  balsam  which 
has  been  softened  with  turpentine  till  its  index  is  precisely  the  same  as  that  of 
crown  glass,  1*525,  and  covered  with  a  thin  glass  1-100  inch  thick.  When  it  is 
strongly  illuminated,  rays  will  proceed  from  it  in  every  direction.  The  rays  will 
proceed  through  the  balsam  and  cover-glass,  without  refraction,  to  the  upper  side 
of  the  cover,  whence  those  which  reach  the  cover  at  right  angles  to  its  upper  sur- 
face will  proceed,  still  without  refraction,  while  the  oblique  rays  will  be  refracted  and 
rendered  more  oblique  (Fig.  13).  If  the  emergence  is  into  air,  the  ray  at  400  will  be 
so  much  refracted  as  to  emerge  nearly  parallel  to  the  surface  of  the  glass,  while 
those  of  41 0  and  more  will  have  no  emergence  at  all,  but  will  be  totally  reflected. 
It  is  evident,  then,  even  for  a  lens  of  nearly  1800,  if  dry,  a  balsam  or  glass  angle  of 
82 0  is  beyond  the  utmost  limit,  for  even  if  the  front  of  the  lens  comes  as  close  to 
the  cover  as  possible,  without  absolute  contact,  and  so  takes  in  rays  of  the  greatest 
possible  divergence  in  air,  there  will  still  be  rays  beyond  which,  having  no-  emer- 
gence into  air,  can  not  reach  the  lens  at  all.  Suppose,  however,  that  the  thin  film 
of  air  between  the  cover-glass  and  the  front  of  lens  is  replaced  by  one  of  water,  or, 
still  better,  of  glycerine.  What  then  will  happen  ?  Why,  the  rays  which  suffered 
so  much  refraction  upon  their  emergence  from  glass  into  air,  will  suffer  much  less 
on  emergence  from  glass  into  glycerine.  The  ray  at  400,  which  was  refracted  to 
780  29' — on  emergence  into  air  is  now  only  refracted  to  41 0  40' — and  the  rays  be- 
tween 41 0  and  750,  which  were  totally  reflected  at  the  upper  surface  of  cover, 
have  now  an  emergence  into  glycerine,  and  part  of  them,  at  least,  become  available 
(Fig.  14). 

I  now  have  here  a  duplex  1-6,  made  by  Mr.  R.  B.  Tolles,  and  marked  Bal- 
sam Angle  950,  Air  Angle  1800 — the  identical  label  which  so  excited  Mr.  Wen- 
ham's  horror  and  amazement.  It  is  an  immersion  lens,  and  works  well  when 
immersed  in  glycerine.  We  will  use  it  over  the  same  slit  slide  that  we  tested  the 
1-5  on.  We  find  that  over  this  comparatively  thick  cover  we  still  have  full  "003 
working  distance,  which  is  ample.  Our  candle  being  in  place  again  we  turn  it  around 
the  object  as  a  centre  till  we  find  the  field  darkened,  and  on  looking  at  the  index 


OBJECTIVES  FOR  THE  MICROSCOPE. 


17 


•we  find  the  angle  reads  780,  a  total  air  angle  of  1560.  A  moment's  inspection, 
however,  serves  to  show  that  the  field  is  because  the  light  from  the  candle  does 
not  reach  the  object,  which  is  eclipsed  by  the  shadow  of  the  stage.  It  is  evident, 
then,  that  we  can  not  determine  the  total  air  angle  of  this  objective  by  this  method 
on  this  stage,  though  it  is  much  thinner  and  admits  much  more  oblique  rays  than 
most  ordinary  stages. 

We  know,  however,  that  the  angle  within  the  glass  slide  is  much  less  than 
that  in  air,  and  always  bears  a  simple  ratio  to,  or,  more  accurately,  that  the  sines 
of  the  angles  made  with  the  normal,  by  a  ray  passing  obliquely  from  glass  to  air, 
or  vice  versa,  bear  a  constant  ratio  to  each  other.     If,  then,  we  can  cancel  the 
effect  of  the  lower  surface  of  the  slide,  and  let  the  ray  pass  into  the  slj^K  \^it^ouf\ 
refraction  at  the  lower  side,  we  can  measure  the  glass  angle,  and  frptn^liat  obt^%, 
by  a  simple  application  of  the  law  of  sines,  the  corresponding  aii^^ngle,  unless  the  i 
glass  angle  should  be  larger  than  that  corresponding  to  air  anglp-W  q6£vin  whnply' 
case  that  would  be  indicated.  I     >  aoV^ 

For  the  purpose  of  obtaining  the  glass  angle  of  this  lens  oji^akcjsfimg  the 
effect  of  the  lower  surface  of  the  slide,  and  suffering  the  light  to  pass  into  the  slide 
without  refraction,  I  shall  make  use  of  a  modification  of  an  ingenious  piece  of  ap- 
paratus devised  by  Mr.  Tolles,  and  described  by  him  in  the  Monthly  Microscopical 
Journal  for  July,  187 1,  and  which  Mr.  Wenham  first  called  a  "  wretched  adapta- 
tion," and  afterwards  adopted  (see  Mo?ithly  Mia'oscopical  Journal,  March,  1874, 
page  117).  It  consists  simply  of  a  plano-convex  lens  of  such  thickness  that  when 
the  plane  side  of  the  lens  is  connected  with  the  under  surface  of  the  slide  by  water, 
glycerine,  or  balsam,  the  thickness  of  lens,  balsam,  and  slide  shall  equal  the  radius 
of  curvature.  The  object  on  the  upper  surface  of  the  slide  can  then  be  placed  in 
the  centre  of  curvature  of  the  lens,  and  any  ray  reaching  it  from  the  curved  sur- 
face must  pass  in  the  direction  of  a  radius  of  curvature,  and  consequently  be 
normal  to  the  curved  surface  at  the  point  of  entrance,  and  consequently  again  can 
undergo  no  refraction  there,  but  must  pass  on  to  the  object  in  a  straight  line  from 
the  source  of  light.  If,  now,  the  angle  that  this  ray  makes  with  the  optic  axis  of 
the  instrument  be  measured,  we  get  the  angle  of  deviation  in  glass,  from  which  we 
can  calculate  the  corresponding  angle  in  air,  if  the  glass  angle  be  41 0  or  less  for 
the  semi  aperture,  a  glass  aperture  of  82  °,  or  very  nearly  that,  being  equal  to  1800, 
or  infinitely  near  that  in  air. 

In  this  case  my  hemispherical  lens  is  of  crown  glass ;  index  of  refraction  (mean) 
1*525  ;  radius  of  curvature  0-45  inch ;  thickness  0*33,  leaving  0*12  for  thickness  of 
slide  and  immersion  connection. 

We  will  make  the  connection  with  a  drop  of  soft  balsam,  the  index  of  which 
is  very  closely  the  same  as  that  of  the  glass,  and  mounting  our  candle  as  before, 
swing  it  round  till  we  find  the  field  obscured.    It  does  not  get  to  780  now,  but  stops 


iS 


ON  ANGULAR  APERTURE  OF 


at  500  (Fig.  15),  indicating  a  glass  angle  of  ioo°  for  the  lens  ;*  but  as  less  than  820  of 
glass  angle  is  equal  to  infinitely  near  1800  in  air,  I  have  demonstrated  that  the  lens 
has  an  air  angle  of  1800,  or  infinitely  near  that,  and  admits  rays  which  could  by 
no  possibility  enter  a  dry  lens,  as  they  would,  if  the  object  were  mounted  in  bal- 
sam, be  totally  reflected  at  the  upper  surface  of  the  cover ;  or,  if  it  was  a  dry  mount, 
at  the  upper  surface  of  the  slide.  To  prove  this  it  is  only  necessary  to  move  into 
the  field  that  part  of  the  slit  where  the  balsam  stops  and  the  dry  mount  begins. 
The  balsam  mounted  part  is  brilliantly  illuminated,  but  the  dry  part  is  in  darkness 
(Fig.  16)  till  the  light  is  turned  back  to  about  400  from  axis,  when  the  ray,  being 
within  the  critical  angle  from  glass  to  air,  passes  through,  and  the  dry  part  of  the 
slit  becomes  illuminated.  We  have  here,  then,  two  objectives  ;  a  dry  %  and  an 
immersion  1-6. 

The  most  oblique  ray  which  the  dry  lens  can  utilize,  makes,  in  passing  through 
the  cover-glass,  an  angle  of  32 0  30'  with  the  optic  axis  of  the  objective,  and  has 
an  emergence  into  air  at  550  from  said  axis,  thus  giving  this  objective  65 0  of  glass 
angle,  or  no°  of  air  angle. 

The  most  oblique  ray  which  the  immersion  1-6  can  utilize,  makes,  in  passing 
through  the  cover-glass,  an  angle  of  500  with  the  optic  axis  of  the  objective ; 
and  on  account  of  its  great  obliquity,  can  have  no  emergence  into  air,  but 
emerges  into  glycerine  at  520  23' — thus  giving  this  objective  a  glass  angle  of  ioo°, 
a  glycerine  angle  of  1040  46',  and  an  air  angle  of  (infinitely  near)  1800.  That  is 
to  say,  this  lens  can  take  up  and  utilize  every  ray  which,  radiating  from  a  balsam 
mounted  object,  could  possibly  have  emergence  into  air;  and  can  also  receive 
and  utilize,  when  immersed  in  glycerine,  a  goodly  pencil  of  rays  which  could  never 
have  emergence  into  air  at  all.    (Fig.  17). 

Will  any  man  in  his  senses  venture  to  say  that  the  dry  lens  has  the  larger 
aperture  of  the  two?  I  think  not;  and  yet  that  is  just  the  result  to  which  we 
must  come  if  we  take  the  isosceles  triangle  method  of  measurement  devised  by  Mr. 
Wenham,  and  endorsed  by  President  Brooke.    The  following  are  the  elements : 


For  the  dry  — 

Clear  aperture  of  front,  -  -  -  -077 

Distance  of  focal  point,  -  -  '019 

Vertical  angle,      -  -  -  -  i27°28' 

For  the  immersion  1-6 — 

Clear  aperture  of  front,  -  -  -  -031 

Distance  of  focal  point,  -  -  '013 

Vertical  angle,       ...  -  ioo° 

or,  270  28'  less  than  that  of  the  dry 


Figure  18  shows  the  two  triangles  superimposed,  that  of  the  dry  lens  being 


*  The  glass  (or  b;ilsam)  angle  of  this  lens,  when  adjusted  for  best  definition  over  Moller's  balsam-mounted 
probe-platte,  is  05°.  The  thirker  cover  used  in  this  experiment  necessitating  considerable  closing  of  the  com- 
binations, and  (in  this  lens)  increase  of  aperture. 


OBJECTIVES   FOR  THE  MICROSCOPE. 


in  dotted  lines ;  and  I  ask,  in  all  seriousness,  if  anything  further  is  needed  to 
demonstrate  the  utter  absurdity  of  the  statement  that,  "  If  an  isosceles  triangle  be 
described,  the  base  of  which  is  ten  times  the  measured  diameter  of  the  front  lens, 
and  the  altitude  ten  times  the  measured  distance  of  the  focal  point  from  the  same 
surface,  the  vertical  angle  of  that  triangle  will  correctly  represent  the  maximum 
available  aperture." 

I  think,  then,  that  I  have  demonstrated  that :  The  angular  aperture  of  an  objective 
is  the  angular  difference  between  the  most  oblique  rays  radiating  from  an  object 
which  the  lens  can  gather  up  and  bring  to  a  common  aplanatic  focus.  That,  as  the 
obliquity  of  these  rays  will  differ  in  the  same  objective,  according  as  it  receives 
them  from  air,  water,  glycerine,  or  balsam,  and  there  is  but  one  part  of  the  course 
common  to  all  cases,  and  that  is  in  the  glass  cover  or  slide,  that  this  is  the  angle 
which  should  be  determined  by  measurement,  and  the  others  calculated  from  it.* 

I  also  believe  that  I  have  demonstrated  that  a  lens  may  have  an  air  angle  of 
(infinitely  near)  1800,  and  a  glass  angle  still  wider  than  that  corresponding  to  in- 
finitely near  1800  air. 

But  here  the  old,  old  question  of  cut  bono,  What  is  the  good  of  this  enormous 
aperture  ?  may  fairly  come  up ;  and  I  reply,  that  it  being  the  function  of  a  micro- 
scope objective  "  to  gather  up  otherwise  diffused  and  lost  rays  of  light  that  issue  from 
an  object,  and  bring  the  whole  rescued  bundle  of  rays  to  a  focus,  in  which  the  image 
of  the  source  from  which  they  stream  is  brilliantly  reproduced,"  then  it  may 
fairly  be  inferred  that  the  objective  which  can  gather  up  and  bring  to  a  focus  the 
most  of  these  otherwise  lost  rays,  theoretically  at  least,  is  the  best  objective. 

And  here  theory  and  practice  go  hand  in  hand.  It  is  a  task  of  immense  dif- 
ficulty to  construct  an  objective  which  will  gather  up  and  bring  to  a  common  focus 
these  extremely  wide  pencils,  but  when  it  is  done  by  the  hand  of  a  master  the 
result  is  splendid. 

Minute  details  of  structure,  invisible  with  lenses  of  equal  or  greater  amplifying 
power  but  smaller  aperture,  are  clearly  revealed.  The  images  are  at  once  sharper, 
clearer,  and  brighter.  So  much  so  that  they  will  bear  examination  with  extremely 
deep  eye-pieces,  and  actually  more  amplification  and  better  definition  can  be  ob- 
tained with  a  1-6  of  1800,  than  with  a  1-16,  1-25,  or  1-50  of  say  1400.  These  lenses 
are  then  economical.  The  owner  of  a  1-6  of  1800  (if  as  thoroughly  corrected  as  this 
one)  has  no  need  to  purchase  a  1-16  or  1-25  ;  by  a  change  of  eye-piece  he  can  get 
all  the  amplification,  definition,  and  resolution,  that  the  shorter  focus  objectives  would 
give  him,  with  larger  field  and  longer  working  distance.  I  have  compared  this  1-6 
with  a  splendid  1-50.  The  work  of  the  1-6  was  unquestionably  superior,  and  with 
it  I  can  work  through  covers  1-100  of  an  inch  thick;  while  for  the  1-50  extra 
thin  covers  had  to  be  specially  imported. 

*  This  is  practically  the  idea  advanced  by  Prof.  Abbe,  of  Jena,  in  his  system  of  numerical  apertures,  though  I 
had  never  heard  of  his  plan  when  this  paper  was  written. 


/ 


NOTE. 


In  making  measurements  of  aperture  by  the  method  here  proposed,  it  is  necessary  that 
the  experiments  be  conducted  in  a  dark  room,  where  the  toy  candle  on  the  microscope  is  the 
only  source  of  light.  If  other  sources  of  light  are  present,  they  are  sure  to  confuse  the 
measurement,  not  only  by  the  introduction  of  stray  rays  into  the  objective,  but  also  by  their 
effect  upon  the  retina,  preventing  the  recognition  of  perfect  definition  of  the  object  when 
illuminated  by  the  feeble  light  of  a  toy  candle. 


APPENDIX. 


As  the  index  of  refraction  of  any  medium  differs  for  different  parts  of  the  spectrum, 
and  the  index  for  the  same  part  of  the  spectrum  will  be  found  to  vary  somewhat  in  different 
specimens  of  the  same  medium,  I  have  thought  it  best  to  give,  as  an  appendix,  the  several 
indices  ot  refraction  made  use  of  in  this  paper. 

It  has  not  seemed  desirable  to  carry  these  indices  beyond  three  places  of  decimals  in  any 
case,  and  thus  it  happens  that  the  mean  index  of  Air,  which  is  usually  given  as  1*000294,  is 
taken  as  unity  (1*000).  The  index  of  crown  glass  (1*525),  is  lower  than  that  usually  given  in 
works  on  Optics,  but  is  that  of  the  crown  glass  actually  used  by  Mr.  Tolles,  in  his  objectives. 
The  relative  indices  I  have  calculated  myself  from  these  data. 

Of  course  all  the  indices  given  are  only  "  means,"  and  are  for  the  deviation  of  the  "  Green 
Ray,"  (Frauenhofer's  line  E). 

TABLE  OF  POSITIVE  AND  KELATTVE  TNDICES  OF  KEFKACTION. 

Means  for  Green  Eay— Frauenhofer's  Line  E.      .  •-' 


«™     :  ■  • 

Air   1*000 

Water   1*336 

Glycerine  -  1*475 

Balsam  (thinned  with  Turpentine)  --------      -  1*525 

Crown  Glass  -      -  -      -      -      -  1*525 

Relative  Indices. 

Water  to  Air   0*749 

"       "     Crown  Glass  or  Balsam    -        -        -        -        -        -  1*142 

"     Glycerine   1*102 

Crown  Glass  or  Balsam  to  Air         ------  o*656 

Water   0*876 

Glycerine   0*967 

Glycerine  to  Air   0*678 

Water       -   0*958 

Crown  Glass  or  Balsam       ------  1-034 


The  Positive  Index  for  Air  being  taken  as  unity,  the  Relative  Indices  f romAir  to  the  other 
media  will,  of  course,  correspond  with  their  Positive  Indices  as  above. 


FlclURE  1. 

Spherical  Aberration. 
Depth  <>f  Focus  or  Penetration. 
Not  Drawn  to  Scale. 


Geo.  E.  Blackham.  M.S.,  Del. 


FlGT'BE  2. 

Scale  X  100. 


Geo.  E.  Slaekhtm,  m. 


FlGTTRE  3. 

Scale  XlUO. 


(ten.  E.  Blaekham.  M.S..  Del. 


Figure  L 


Scale  X 100. 
Reduction  of  Aperture  iiy  Setting. 


A.  B.  Diameter  of  Front,  -08 

C.  D.  Radius  <  tf  Curvature,  0-15 
A.  H.  &  B.  I.  Allowance  tor  setting. 

H.I.  Available  Diameter  ol  Front.  -028 

C.  G.  Frontal  Distance.  11157 

D.  G.  Anterior  Conjugate  Focus,  -Q307 


C.  Centre  of  Curvature. 

F.  Principal  Foots. 

H.  G.  I.  Angle  of  Aperture  83°  26'. 

Posterior  ConjugateFocusinot  shown)  10  ins. 


Geo.  K  WarJehnm.  M.D.,  Del 


Figure  5. 

mb.  wenham's  apparatus  for  excluding  all  but  the  central  pencil. 
Figure  enlarged  two  diameters  from  M.  M.  .1..  Dec,  1876. 


a 


a.  Working  diameter  of  Objeet-glass. 
ft.  Central  pencil. 

c.  c.  Oblique  or  lateral  pencils  enclosing  the  field  of  view. 

d.  d.  Slit  of  considerable  width  attached  to, 
e.  Glass  slip. 


Geo.  E.  mackliam,  M.D..  Del. 


Figure  6. 


Scale  X  20. 

Tolles'  Student  4,  Dry,  Uncovered  Object. 

A.  B.  Diameter  of  Front,   O'lti  inch. 
C.  F.  Frontal  Distance,      0-015  " 

Resulting  Angle  of  Aperture  (a  la  Wenham)  158°  46'. 


i 

A  c 

 B 

 ' 

I  / 

1  / 
*  / 
%  / 

NORMA  L 

Gro.  E.  Blackham,  M.D.,  Del. 


Figure  8. 

Scale  X  20. 

Tolles'  Student  4,  Dry,  Uncovered  Object. 

C.  D.  Diameter  of  Clear  Aperture  of  Front,  -077. 
Frontal  Distance  Uncovered,  -015. 

Resulting  Angular  Aperture  (a  la  Wenhain),  137°  26'. 


C  '0 

77  D 

£10.1 

1  / 

1  / 
B  / 

5  /$> 

i  A 
I  / 

i  / 

^^^^^^ 

Geo.  E.  Blackham,  M.D..  Del. 


FlGTTKE.  9. 

Scale  X  20. 

Tolles'  Student  i,  Dry,  Uncovered  Object. 

Angle  of  Aperture,  actual,  by  measurement,  100°. 

Frontal  Distance,  015. 
A.  B.  Resulting  Diameter  of  Front  actually  utilised,  -0358. 


Geo.  E.  Blackham,  M.U.,  Del 


FlGTTKE  10. 


Scale  X  20. 

Tolles'  Student  i,  Dry,  Covered  Dry  Mount. 

Clear  Aperture  of  Front,  -077. 
Working  Distance,  -009. 
Thickness  of  Cover  and  Air  Space,  '010. 
Total  Frontal  Distance,  019. 

Resulting  Angular  Aperture  (u  la  Wenhain),  127°  : 


Geo.  K  BlacMam,  M.S.,  Del. 


Figure  11. 

Tollea'  Student  I  Dry,  Covered  Dry  Mount. 

Diameter  of  Exposed  Front,  016. 
Working  Distance,  0-009. 
Thic  kness  ot  Cover  and  Air  Space,  0-010. 
Total  Frontal  Distance,  0-019. 

Resulting  Angular  Aperture  [a  lit  Weuham),  153°  16', 


AIR 


f 


Geo.  E.  Blackham.  M.D..  Del. 


Figure  12. 


Scale  X  20. 

Tolles'  Student  I,  Dry,  Coverejl  Dry  Mount. 

Working  Distance,  '009  inch. 

Thickness  .  if  Cover  and  Air  Space,  -010  " 
Total  Frontal  Distance.  -019  " 

Air  Angle  (by  measnreraent),  110°. 

Glass  "  ",       "  65°. 

Resulting  Diameter  of  Front  actually  utilised,  '0381  indies. 


Geo.  E.  Blaekham.  M  R.  Del. 


7 


Figure  13. 


Geo.  E.  Blackham,  M.S..  Del 


Figure  14. 

Balsam  Mount. 

Effraction  of  Kays  from  Crown  Glass,  index  l-525. 
To  Glycerine,  "  1H75. 

The  dotted  lines  show  the  paths  the  rays  -would  take  if  the  Glycerine  were  replaced  by  Air. 


\  s 
\  * 

\3T  * 

_?<9°2S'          \  Glyc 

 KTiar         —   \ 

ERINE   /              78°  29'.   ~~~~~~ 

1  A 

i  / 

•8  / 
§  / 

! 

Am  \ 

Geo.  E.  Blackham,  M.S.,  Del 


Figure  15. 
Balsam  Mount. 
Scale  X  5. 


Geo.  E.  Blackliam.  M.I)..  Del. 


Figure  16. 
Scale  X  20. 


Geo.  E.  Blaekliam,  M.D..  Del. 


I 

Figure  17. 
Balsam  Mount. 
Scale  X  20. 


Refraction  of  Rays  passing  from  Crown  Glass  to  Glycerine. 
The  dotted  lines  represent  the  course  the  Rays  would  take  if  the  Glycerine  were  replaced  by  Air. 


Geo.  E.  Jilackham,  M.D.,  Bel. 


Scale  X  40. 


Comparison  of  Wonham's  Triangles  f. >r  T< 'lies'  Dry  i  of  110°  Air  Aperture,  and  Tolles'  Wet 
1-0  of  180°  Air  Aperture  (Glycerine  Immersion). 


Both  in  use  over  Balsam  Mount  ( '<  iver  'ill  inch  llii.-k. 


A.  B.  C.  Triangle  for  Dry  j. 

A.  B.  Clear  Aperture  of  Front,      '077  inch. 

F.  C.  Frontal  Distance,  "019  " 

A.  C.  B.  Angular  Ap.  (a  la  Wenham),  127°  28'. 


D.  E.  C.  Tr 
D.  E.  Cl( 
G.  C.  Fi- 


nale for  Wet  1-6. 
r  Aperture  of  Front, 
ital  I  >i  stance, 


•031  inch. 
■013  " 


D.  C.  E.  Angular  Ap.  (a  la  Wenharn),  100° 


10  inch  thi 

own  Glass  0 

Slide  of  Or 

NORMAL 

Am 

Geo.  E.  Blackham,  M.D..  Del. 


